78 D. Day et al. Here, the error is calculated from the Euclidean or L2 normof ¨uh −P¨uH, and Pis a fine to coarse mesh transfer linear operator called the prolongation. The solution of the optimization problem is the maximum of the solutions over each element. Eigenvalues connect accelerations and displacements, ¨q =− 2 h q and ¨b =− 2 H b and rigid body modes can be ignored. Due to q =− −2 h ¨ q, there holds σv h 2 = ¨ qTX¨q for X= χij , χij = 3 2 Ψdev ω2 : Ψdev ω2 (8.11) Here, X is a symmetric matrix with a positive diagonal, composed of the deviatoric stress modes and eigenvalues. The bound on the stress is determined by solving the nonlinear Eq. (8.10). Some assumptions are made which explain how, for the purposes of this study, solving the nonlinear equation is relatively easy because several complications do not arise. Details are given in Appendix 1. The basic problem is as follows: for given ¨b on the coarse mesh, to maximize σv h 2 = ¨qTX¨q over all ¨q such that the error in acceleration, ¨uh −P¨uH 2 L2 < δ 2. This provides an understanding of the relationship between the accuracy in acceleration to the accuracy in stress, and how this bound varies with the state of acceleration on the coarse mesh, ¨uH. 8.3 Analysis Examples FEA was performed for three different example problems and a stress-acceleration error relationship was determined for each case. The error is defined as the percent difference in stress or acceleration from a base model that ideally represents the system. Using the theory presented in the previous section, the first case examines the error bound for an unsupported, or free-free, beam. An acceleration error is imposed and the relation to the maximum modal von Mises stress error is found. The second and third cases involve more practical examples, in which random vibration of a cantilever beam and two-material, joint rectangular beam is considered. Material properties and beam geometry are varied (perturbed) to obtain an error in acceleration that is related to the VRMS error. All analysis was performed using Sandia’s in-house finite element code Sierra/SD, along with MATLAB and Python scripts. 8.3.1 Case 1: Free–Free Beam Analysis for this problem follows the steps outlined in Sect. 8.2.1, with further details provided in Appendix 1. An acceleration is initially applied to all nodes in the coarse mesh and the resulting L2 error in acceleration is linearly varied from an initial δmin, toδmin +2 c , which is twice the norm of the initial fine mesh acceleration. For the free–free beam, the maximum modal von Mises stress was calculated on the fine mesh for different values of this L2 acceleration error. Stress and displacement mode shapes from 16 calculated modes were used. The beam material is steel and it is 1 in. (25.4 mm) in diameter and 10 in. (254 mm) in length. The beam was meshed with a total of 1920 8-node, linear hexahedral elements (Hex8) as shown in Fig. 8.1. All stress calculations were done on the fine mesh, with a coarse mesh of 240 elements used to apply the acceleration, as mentioned in Sect. 8.2.1. The resulting plot in Fig. 8.2 shows a linear relationship between the maximum stress error in the model and the acceleration error. The modal von Mises stress distributions1 for the initial and final stress states are also presented. The acceleration error reached a maximum of 393%, and the error in the von Mises stress was 154%, giving the maximum stress error to acceleration error ratio of 0.392. This error ratio, defined as the maximum percent VRMS error divided by the maximum acceleration percent error relationship provides a measure of the sensitivity of stress error change relative to the acceleration error. It also describes how the errors are bound and is the slope of the percent error plot for a linear relation. This relation between the maximum modal stress in the model and the acceleration error is a first step to obtaining a practical error bound. The example case is theoretical, since the accelerations are ideally applied and the resulting modal stresses cannot be used to directly evaluate failure. We are typically interested in more realistic boundary conditions and 1Like the mode shape amplitudes, the mode shape stresses can be arbitrarily scaled, so only the relative values are important here.
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